The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part Bl. Beijing 2008 
213 
Unsupervised learning will be used in this thesis to detect echo 
pulses. This is done by fitting Gaussians to the waveforms. It 
will be assumed that the waveforms were generated from a 
distribution which is the sum of simpler distributions. That is, 
the samples of the waveforms are assumed to arise from the 
following distribution. 
f{x) = Y d P j 
i-1 
fj{x)<=N(n n <j]) 
Where k is the number of Gaussians, fj (x) is the Gaussian 
probability density function, p ■ is the relative weight 
of fj (x), JUj is the expected value and G j is the standard 
deviation of the fitted Gaussians. 
The EM algorithm, which will be used to fit the Gaussians to 
the waveforms, is a widely used approach in learning the 
presence of unobserved variables. Original formula of EM 
algorithm: 
ß, = . 
PjfMi) 
(l) 
Y" Q 
(2) 
n 
Y" O i 
(3) 
P 
J x n 
CT. = 1 
(4) 
J \ 
PjXn 
Qj is the probability that sample i belongs to component j and 
k is the number of components that are fitted to the 
waveforms .For each component j, the mean value fl • and 
standard deviation cr . are estimated. The mean value JUj 
will be used as the position of the echo and the standard 
deviation <J. as the width of the echo. The likelihood 
estimates for ¡d - and <T y are found by iterating through 
formula (l)-(4). The algorithm needs to be initialized with start 
values. 
In the estimation step of the EM algorithm, the expected value 
of each hidden variable is calculated assuming that the current 
hypothesis holds.. In the maximization step (2)-(4), a new 
maximum likelihood hypothesis is calculated assuming that the 
value taken on by each hidden variable is its expected value 
calculated in the estimation step. The hypothesis is replaced by 
the new hypothesis and a new iteration is made. However, 
original formula of EM algorithm does not take into account the 
intensity of sample i. consequence of the improved EM 
algorithm: 
Formula (2) insert in formula (3)-(4): 
i:„e,«-*,)’ 
nx Yj" M Q^- pf 
'E.Av-^y <6) 
i 
Now, intensity N insert in numerator and denominator: 
ZLW (7) 
HAQ, 
(8) 
i Urn 
Now, formula (7)-(8) revert to form of EM algorithm: 
Pjfj( 0 
Qu = 
Pi = 
Hj.iPjfj® 
ILw 
nx Pj x 
H 
HWV-m,) 1 
(9) 
(10) 
(11) 
(12) 
nx Pi x Y^ N i 
S is the number of samples in the waveform and N i is the 
intensity for sample i. 
4. EXPERIMENTS AND DISCUSSION 
4.1 SLICER 
The SLICER .dat data files are derived from the standard Laser 
Altimeter Processing Facility .geo files which include 
geolocation results and instrument data. The inclination and 
azimuth of the transmit pulse are also derived from the .geo roll, 
pitch and yaw data, simplifying correction for waveform slant 
range distances. The diameter of each laser footprint, based on 
an approximate laser divergence and the ranging distance, is 
also provided. 
4.1.1 Pre-processing 
Figure 4 and Figure 5, The waveforms have to be thresholded to 
remove noise before the EM algorithm is applied. The threshold 
is derived on a per shot basis by establishing the mean 
background noise occurring in the waveform record, the last 5% 
of the waveform record is used to calculate threshold <y . . 
noise 
All samples of the waveform below the threshold cr . are 
~ noise 
then set to zero.